Optimal. Leaf size=18 \[ \frac{x}{2}-\frac{\sinh (x)}{2}+\frac{\cosh (x)}{2} \]
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Rubi [A] time = 0.0253834, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3131} \[ \frac{x}{2}-\frac{\sinh (x)}{2}+\frac{\cosh (x)}{2} \]
Antiderivative was successfully verified.
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Rule 3131
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{1+\cosh (x)+\sinh (x)} \, dx &=\frac{x}{2}+\frac{\cosh (x)}{2}-\frac{\sinh (x)}{2}\\ \end{align*}
Mathematica [A] time = 0.039632, size = 18, normalized size = 1. \[ \frac{x}{2}-\frac{\sinh (x)}{2}+\frac{\cosh (x)}{2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 28, normalized size = 1.6 \begin{align*} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16206, size = 14, normalized size = 0.78 \begin{align*} \frac{1}{2} \, x + \frac{1}{2} \, e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.438, size = 72, normalized size = 4. \begin{align*} \frac{x \cosh \left (x\right ) + x \sinh \left (x\right ) + 1}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.702367, size = 34, normalized size = 1.89 \begin{align*} \frac{x \tanh{\left (\frac{x}{2} \right )}}{2 \tanh{\left (\frac{x}{2} \right )} + 2} + \frac{x}{2 \tanh{\left (\frac{x}{2} \right )} + 2} + \frac{2}{2 \tanh{\left (\frac{x}{2} \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2229, size = 14, normalized size = 0.78 \begin{align*} \frac{1}{2} \, x + \frac{1}{2} \, e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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